Quantum spatial dynamics of high-gain parametric down-conversion accompanied by cascaded up-conversion

Quantum cascaded up-conversion (CUpC) of parametric down conversion (PDC) in a finite nonlinear $\chi^{(2)}$-crystal is studied theoretically within approximation. The exact solution for creation and annihilation operators presented the form Bogoliubov transformation valid high-gain regime explicitly includes non-zero wavevector-mismatch both PDC CUpC. With use characteristic equation amplification oscillating regimes are analysed degenerate, three- four-mode cases. We show that exists under fulfilment phase-matching conditions while CUpC processes separately non-phase-matched. influence on quadrature squeezing degenerate estimated.